| This question, in various forms, does just
keep coming up... While I'm going to take the opportunity to
refer to a detailed, formal, and I hope dispositive paper just published by the
Minnesota group (doi 10.1021/jp205508z), I'll also make some effort here to
be more cookbooky in an explanation here. Pedagogy in front! Not just for
Gaussian, but more general. The free energy of
solvation is, clearly, defined as the DIFFERENCE in the free energy of a species
in the gas phase and in solution. Free energy is an ensemble property, not a
molecular property, so we are immediately faced with the need to make some
approximations in order to render the modeling
tractable. In the gas phase, those approximations are
by now fairly standard and nearly universal. We make the ideal-gas approximation
(so that the partition function of a mole of molecules is simply the product of
Avogadro's number of molecular partition functions), and we make the rigid
rotator and harmonic oscillator approximations to simplify the rovibrational
partition functions, and, voila, we have a means to compute a standard state
free energy (Gibbs free energy -- free enthalpy for my colleagues in (far more
logical) German-speaking countries). Pretty much every electronic structure
program on earth will do this for you when you run a vibrational frequency
calculation in the gas phase. Poof -- you get E, H, S (third law), and G (and Cv
too, if you care). Note that you should be careful to
recognize that (i) unless you overrode it, the program chose some default
standard state concentration (1 bar? 1 atm? you should know...) and temperature
(nearly always 298 K) and scale factor for vibrational frequencies (typically 1,
but that might not be the best value for a given level of theory) and (ii) the
harmonic oscillator approximation is catastrophically bad for super-low
frequency vibrations (below, say, 50 wavenumbers, to pick an arbitrary value).
There are fixes for the latter problem, but I'll let someone else post about
that. Now, what about G of a solute in solution? Well,
to begin, since we're not dealing with a pure component anymore (at least, not
if the solvent is different than the solute), we need to assume that we have an
ideal solution, and we should recognize that we're talking about a partial molar
quantity (more often referred to in the rigorous literature as a chemical
potential than a free energy, but that's a matter of tradition). In any case, we
need to assemble a free energy as a sum of electronic
energy (i.e., potential and kinetic energy of electrons for a fixed set of
nuclear positions) coupling to medium (which includes
electrostatic and non-electrostatic components, although, it being chemistry,
EVERYTHING is electrostatic... -- in practice, however, with continuum models,
electrostatic means what you get by assuming the molecule is a charge
distribution in a cavity embedded in a classical dielectric medium in which case
one can apply the Poisson equation -- non-electrostatic is everything else --
dispersion, cavitation, covalent components of hydrogen bonding, hydrophobic
effects, you name it) temperature dependent
translational (?), rotational (?), vibrational, and conformational contributions
-- the question marks indicate conceptual issues So, a
few points to bear in mind. The optimal geometry in solution is unlikely to be
the same as that in the gas phase -- but it might be close. You just have to
decide for yourself if you want to reoptimize or
not. Same for the vibrational and conformational
contributions to free energy in solution -- they might be very, very close to
those in the gas phase -- or they might not. If you assume that they ARE the
same, you avoid having to do a frequency calculation in solution (and you avoid
wondering what it means to do vibrational frequencies in a continuum, which in
principle means a surrounding that is in equilibrium with the solute -- but how
can a medium composed of molecules be fully in equilibrium with a molecular
solute on the timescale of the solute's vibrations, since the solvent vibrations
are on the same timescale?) Note that if I assume no
change in the various parts of the free energy EXCEPT for the electronic energy
and solute-solvent coupling (more on that momentarily), life is pretty easy. The
free energy of solvation is the difference in the self-consistent reaction-field
(SCRF) energy INCLUDING non-electrostatic effects, and the electronic energy in
the gas phase. That is, I look at the expectation value of <H+(1/2)V> for
the solvated wave function, where V is the reaction field operator, add
non-electrostatic effects (typically NOT dependent on the quantum wave function,
so added post facto, although there are a few exceptions in the literature),
subtract the expectation value of <H> for the gas-phase wave function
(note that you might have done the two expectation values at different
geometries, or you might have used the gas-phase geometry for both -- your
choice -- the former is more "physical", certainly, but the latter is a useful
approximation in many instances), and you are done. You've got the free energy
of solvation FOR IDENTICAL STANDARD-STATE CONCENTRATIONS. That is, the number
you have in hand assumes no change in standard-state concentration. However,
many experimental solvation free energies are tabulated for, say, 1 atm gaseous
standard states and 1 M solution standard states. To compare the computed value
to the tabulated value, one needs to correct for the standard-state
concentration difference. In the interest of the
cookbook, let me be more practical. Thus, let's say that I compute a gas-phase G
value, including all contributions, electronic and otherwise
of -3.000 00 a.u. and, let's
say that the electronic energy alone in the gas phase
is -3.020 00 a.u. (so ZPVE and thermal
contributions to G are +0.020 00 a.u.) and, finally,
let's say that my SCRF calculation provides an electronic energy INCLUDING
non-electrostatic effects of -3.030 00
a.u. In that case, my free energy of solvation is
-0.010 00 a.u. (difference of -3.030 00 and -3.020 00 a.u.) And, if I want to
think about my free energy in solution, I can make the assumption that there is
no change in the ZPVE and thermal contributions, in which case I would have G in
solution equals -3.010 00 a.u. (which is gas-phase G of -3.000 00 a.u. plus free
energy of solvation -0.010 00 a.u.) But, just to be
clear, if my gas phase G referred to a 1 atm standard state concentration, for
an ideal gas at 298 K and 1 atm, that implies a molarity of 1/24.5 M. If I want
my G in solution to be for a 1 M standard state, I need to pay the entropy
penalty to compress my concentration from 1/24.5 M to 1 M, which is about 1.9
kcal/mol (the proof is left to the reader...) So, my 1 M free energy in solution
is not -3.010 00 but rather about -3.006 99 a.u. The
above is an example of how almost all free energies of solvation and free
energies in solution are computed in the literature using continuum solvation
models (at least if they're done properly!) Lots of
important details glossed over a bit above (in the interests of clarity, I
demur). But, to be more thorough, let's note: 1)
Why was it (1/2)V in the SCRF calculation? -- the 1/2 comes from linear
response theory and assumes that you spend precisely half of the favorable
coupling energy organizing the medium so that it provides a favorable reaction
field. 2) How can there be a translational
partition function for a solute in solution? There isn't one -- but there is
something called a liberational free energy associated with accessible volume,
and Ben-Naim showed some time ago that the value is identical to that for the a
particle-in-a-box having the same standard-state concentration -- i.e., there is
no change on going from gas-phase translational partition function to
liberational partition function for the same standard-state concentration for an
ideal solution. When there are issues with non-accessible volume, however,
account must be taked (cf. Flory-Huggins theory). 3)
How can there be a rotational partition function for a solute in solution?
There isn't one -- solute rotations become librations that are almost certainly
intimately coupled with first-solvation-shell motions. In essence, assuming no
change in "rotational partition function" implies assuming no free energetic
consequence associated with moving from rotations to librations. This remains a
poorly resolved question, but, in practice, since most continuum solvation
models are semiempirical in nature (having been parameterized against
experimental data) any actual changes in free energy have been absorbed in the
parameterization as best as possible. If you find that unsatisfying, hey, feel
free not to use continuum solvent models -- it's certainly ok by
me... 4) Where did those non-electrostatic
effects come from? Every model is different in that regard, and I won't attempt
to summarize a review's-worth of material in an email. Lots of nice Chem. Rev.
articles over the years on continuum models if you want to catch
up. Finally, what is described above is a popular
approach for computing solvation free energies and free energies in solution,
but by no means the only approach out there. A non-exhaustive list to compute
either or both solvation free energies or free energies in solution includes
free-energy perturbation from explicit simulations, RISM-based models,
fragment-based models derived > from a statistical mechanical approach
(including COSMO-RS and variations on that theme), fragment-based models from
expert learning, and models relying on alternative physicochemical approaches to
computing interaction energies (e.g., SPARC). These alternative models can be
quantal, classical, or SMILESal (which is to say, more in the realm of
chemoinformatics than physical chemistry). Let a thousand flowers
bloom. I hope that this post serves as a useful
archival reference for CCL users present and future. Best wishes to all for a
peaceful winter solstice (or summer, for my antipodeal
colleagues). Chris On
Nov 30, 2011, at 1:46 PM, Close, David M. CLOSED#,#mail.etsu.edu wrote:
-- Christopher J. Cramer Elmore H. Northey Professor University of Minnesota Department of Chemistry 207 Pleasant St. SE Minneapolis, MN 55455-0431 -------------------------- Phone: (612) 624-0859 || FAX: (612) 626-7541 Mobile: (952) 297-2575 email: cramer|a|umn.edu jabber: cramer|a|jabber.umn.edu (website includes information about the textbook "Essentials of Computational Chemistry: Theories and Models, 2nd Edition") |